1.
A 1 kg mass hangs from a spring with stiffness k = 25 N/m. There is no damping. Find the natural angular frequency ω0.
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Answer:
Use ω0 = sqrt(k/m).
Oscillations & resonance practice problems
In this topic you solve mx'' + cx' + kx = F. The unknown x(t) is a displacement: a mass on a spring, a charge in a circuit, a swing angle. You find the natural frequency, classify the damping, and compute the response to a periodic force. Recognition cue: the equation contains x'' together with x, and the solution swings back and forth or decays. You will practice: Free undamped oscillation, Amplitude and phase form, Three damping cases, Forced oscillation and resonance.
2.
Solve x'' + 9x = 0 with x(0) = 2 and x'(0) = 0.
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Answer:
ω0 = 3. The initial values give C1 = 2 and C2 = 0.
3.
A signal is x(t) = 3 cos 2t + 4 sin 2t. Find its amplitude R.
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Answer:
Use R = sqrt(C1² + C2²).
4.
Solve x'' + 4x = 0 with x(0) = 0 and x'(0) = 6.
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Answer:
ω0 = 2. C1 = x(0) = 0 and C2 = x'(0)/ω0 = 3.
5.
A 4 kg buoy floats in water. When it sits x meters below its rest level, the water pushes it up with force 16x N, so 4x'' + 16x = 0. Find the period T of the bobbing motion.
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Answer:
Divide by 4 to get x'' + 4x = 0.
6.
Find the general solution of x'' + 6x' + 9x = 0.
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Answer:
c² − 4mk = 36 − 36 = 0, so critical damping with double root r = −3.
7.
A model car suspension obeys x'' + x' + 4x = 0. Classify the damping with one word: overdamped, underdamped, or critical.
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Answer:
Here m = 1, c = 1, k = 4.
The result is negative, so the motion is underdamped.
8.
Find the general solution of x'' + 5x' + 4x = 0.
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Answer:
Factor the characteristic equation.
Real roots −1 and −4, so overdamped.
9.
A circuit has a 1 H inductor, a 0.01 F capacitor, and no resistor. The charge obeys q'' + 100q = 0. At t = 0 the charge is q(0) = 0 and the current is q'(0) = 5. Find q(t).
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Answer:
ω0 = 10. C1 = q(0) = 0 and C2 = q'(0)/ω0 = 1/2.
10.
An eardrum model obeys x'' + 4x = 6 cos t. Find a particular solution x_p.
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Answer:
ω0² = 4 and ω² = 1 are not equal, so use the standard formula.
Check: x_p'' + 4x_p = −2 cos t + 8 cos t = 6 cos t.
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More differential equations practice
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